Cavalieri's Method of Indivisibles KIRSTI ANDERSEN Communicated by C. J. SCRIBA & OLAF PEDERSEN BONAVENTURA CAVALIERI(Archivio Fotografico dei Civici Musei, Milano) 292 K. ANDERSEN Contents Introduction ............................... 292 I. The Life and Work of Cavalieri .................... 293 II. Figures .............................. 296 III. "All the lines" . .......................... 300 IV. Other omnes-concepts ........................ 312 V. The foundation of Cavalieri's collective method of indivisibles ...... 315 VI. The basic theorems of Book II of Geometria .............. 321 VII. Application of the theory to conic sections ............... 330 VIII. Generalizations of the omnes-concept ................. 335 IX. The distributive method ....................... 348 X. Reaction to Cavalieri's method and understanding of it ......... 353 Concluding remarks ........................... 363 Acknowledgements ............................ 364 List of symbols ............................. 364 Literature ................................ 365 Introduction CAVALIERI is well known for the method of indivisibles which he created during the third decade of the 17 th century. The ideas underlying this method, however, are generally little known.' This almost paradoxical situation is mainly caused by the fact that authors dealing with the general development of analysis in the 17 th century take CAVALIERI as a natural starting point, but do not discuss his rather special method in detail, because their aim is to trace ideas about infinites- imals. There has even been a tendency to present the foundation of his method in a way which is too simplified to reflect CAVALIERI'S original intentions. The rather vast literature, mainly in Italian, explicitly devoted to CAVALIERI does not add much to a general understanding of CAVALIERI'Smethod of indivi- sibles, because most of these studies either presuppose a knowledge of the method or treat specific aspects of it. Yet there is one apt presentation of CAVALIERI'Stheory of indivisibles; it was published by ENRICO GIUSTI in 1980. Only when this paper was almost finished did I become aware of GIUSTI'S study which is published as an introduction, fuori commercio, to a reprint of CAVALIERI'S Exercitationes geometrieae sex; hence GIUSTI'S and my work overlap. To keep my paper coherent I have not changed the sections which are similar to some in GIUSTI'S book, but only added references to the latter. This procedure is further motivated by the fact that GIUSTI'S approach and mine to CAVALIERI'Smethod are rather different. He gives a background to the understanding of CAVALIERI'S theory and its weakness and does not treat many technical aspects. I intend to supplement the existing literature on CAVA- LIERI with a detailed presentation of his method, including its fundamental ideas, the concepts involved in it, its technique of proofs, and its applications. Further, I try to sketch how mathematicians have understood CAVALIERI'S ideas, Cavalieri's Method of Indivisibles 293 A complete study of the interpretations of CAVALIERI'S theory would be very useful, but requires a paper of its own (a presentation of some of the interpreta- tions can be found in GALUZZI & GUERRAGGIO 1983). Hence I here concentrate on the reactions of some of the 17th-century mathematicians, for example GALILEO, GULDIN, TORRICELLI, ROBERVAL, PASCAL and WALLIS, who were influental in creating an opinion on CAVALIERI'S method; and I add a few examples of 18 th century views on the method. My wish to treat CAVALIERI'S theory in one paper has resulted in another restriction: to leave out the otherwise interesting questions about similarities between CAVALIERI'S ideas and those occurring in medieval and renaissance mathematics. To avoid a confusion between modern concepts and the CAVALIERIan ones I have introduced an ad hoc notation; a list of the symbols employed can be found at the end of the paper. I. The Life and Work of Cavalieri I. 1. Our knowledge of CAVALIERI'S life stems mainly from his letters to GALILEO and other colleagues, from a few official documents, and from his first biographer and pupil URBANO DAVISO, who is not always to be trusted. Since complete bio- graphies of CAVALIERI have been compiled several times from this material, I shall only give a few particulars about his life as background for his mathe- matical work. (For more biographical information, see PlOLA 1840, FAVARO 1888, MASOTTI 1948, CARRUCCIO 1971, and GluSXt 1980.) CAVALIERt was born about 1598 and as a boy in Milan he came in contact with the rather small order of Jesuats, the male section of which was dissolved in 1668. In 1615 CAVAHERI entered this order and on that occasion he probably took the first name BONAVENTUgA. The years 1616-1620, with an interruption of a one-year stay in Florence around 1617 (cf. GIusxI 1980, p. 3), CAVALIZRI spent at the Jesuati convent in Pisa, and he became a mathematical pupil of the Bene- dictine BENEDETTOCASTELLI. CASTELLIwas so satisfied with this student that about 1617 he arranged a contact to his own teacher, GALILEO GALILEI (Cf GALILEI Opere, vol. 12, p. 318). This resulted in more than 100 letters from CAVALIEgI to GALILEO in the period 1619-1641. GALILEO did not answer all of them, but sent an occasional letter to CAVALIERI; of these all but a very few have disappeared. DAWSO'S version of how CAVALIERI took up mathematics is more dramatic than the one presented here. DAVlSO claimed that at the age of twenty-three CAVA- LIERI started an intense study of mathematics, having been told by CASTELLIthat mathematics was an efficacious remedy against depression. However, the facts that in 1617 FEDER1GO BORROMEO asked GALILEO to support CAVALIERI, that in 1618 CAVALIERI temporarily took over CASXELLI'S lectures on mathematics in Pisa, and that CAVALIERI in 1619 applied for a vacant professorship in mathe- matics destroy DAVISO'S thesis of CAVAL~ERI'S late start (GALILEI Opere, vol. 12, p. 320, GIUSTI 1980, pp. 3-4, FAVARO 1888, pp. 4 and 35). 294 K. ANDERSEN CAVALIERI did not obtain the chair in Bologna in 1619, but he went on apply- ing for a lettura in mathematics at various places while moving between Jesuati monasteries in Milan, Lodi, and Parma. In his letters he ventilated the idea that the reason why he was not appointed professor of mathematics was that the Jesuats were not very popular in Rome. Probably through GALILEO'S influence, CAVALIERI eventually obtained a professorship in mathematics at the university of Bologna in 1629. About the same time he also became prior at the Jesuati monastery there. The appointment to the chair of mathematics was only for a period of three years, but CAVALIERI had it renewed until his death in 1647. In his teaching CAVALIERI seems to have followed a three years' cycle of lectures, consisting of comments on EUCLID, Theoriea planetarum, and PTOLEMY'S astronomy (FAVARO 1888, p. 22). Besides his two major works, which will be presented shortly, CAVALIERI published eight books on mathematics and mathe- matical sciences and a table of logarithms. One of the books treating astrology was published under the pseudonym SILVIO FILOMANTIO. The other books were mainly textbooks, and although some of them contain references to results ob- tained by the method of indivisibles they do not deal with the method. Therefore I shall not discuss them further, but refer to the bibliography where their titles are listed. 1.2. The book which made CAVALIERI famous in mathematical circles was Geo- metria indivisibilibus continuorum nova quadam ratione promota, Bologna 1635 (Geometry, advanced in a new way by the indivisibles of the continua); I shall abbreviate this impressive title to Geometria. It is difficult to follow CAVALIERI through the almost 700 pages of this book, so difficult that MAXIMILIEN MARIE suggested that if a prize existed for the most unreadable book, it should be awarded to CAVALIERI for Geometria (MARIE 1883-1888, VO1. 4, p. 90); further, the mathe- matical language CAVALIERI employed in Geometria was characterized by CARL B. BOYER as "confusingly obscure" (BOYER 1941, p. 85). Nevertheless, Geometria was in its time considered so important that it was reprinted in 1653 in an edition which, unlike the first, is paginated continuously. The main reason why Geometria attracted attention was doubtless that most mathematicians of the 17th century were interested in its topic, quadratures and cubatures, and that the number of publications on this subject was small. The mathematicians who carefully studied Geometria were probably few, but neverthe- less it remained a well known book.This is reflected in the treatment of Geometria in general works and articles on 17tU-century mathematics: it is mentioned, but its content is not thoroughly described. That Geometria is still considered an important contribution to mathematics can be seen from the fact that it has been translated into modern languages. Thus in 1940 parts of Geometria with elaborate comments appeared in a Russian edi- tion made by S. J. LUR'E (my inability to read Russian has prevented me from consulting this edition). A complete translation of Geometria into Italian, con- taining many clarifying comments, was published in 1966 by LucIo LOMBARDO- RADICE. Shortly before his death CAVALIERI published another work on indivisibles, the Exercitationes geometricae sex (1647, 543 pp.); this book has received much Cavalieri's Method of Indivisibles 295 less attention than Geometria, but is still mentioned in expositions of CAVALIERI'S contributions to mathematics. My presentation of CAVALIERI'S method of in- divisibles is based on features both from Geometria and Exercitationes, and since I am not going to discuss all his results I shall briefly outline the contents of these two books. 1.3. Geometria consists of seven books. In the first, CAVALIERI clarifies some of his assumptions concerning plane and solid figures. In Book II he introduces the method of indivisibles, or rather his first method which I term the collective method, and proves some general theorems concerning collections of indivisibles.